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March 9th, 2019, 06:13 AM   #1
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Finding square root of matrix without eigenvector

A is a 2x2 matrix.

(0 -2)
(2 0)

Find the square root of this matrix.

I managed to find one of the square root, because I know A is a 90-degree anti-clockwise rotation stretched by a factor of 2.

So I just have to "half" that operation and I get
(1 -1)
(1 1)

However, there is another matrix which is the square root of A and I'm struggling to find it. How can I find it without using eigenvectors/eigenvalues?
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March 9th, 2019, 08:44 AM   #2
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what if we do two rotations the other direction?

$A = 2 Rot\left(\dfrac \pi 2\right)$

$\sqrt{A} = \sqrt{2} Rot\left(\dfrac{-3\pi}{4}\right)$

$\sqrt{A} = \sqrt{2}\begin{pmatrix}-1 &1\\-1 &-1\end{pmatrix}$
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March 11th, 2019, 04:05 AM   #3
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Thank you romsek.
I managed to solve it with your advice and got
(-1 1)
(-1 -1)
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